Fundamentals of Wavelets
Jizheng Di · Medical Entomology and Zoology · 2012
Contents 1 Mathematical preliminaries Some mathematical concepts and knowledge Lp(R) Functions in Lp(R) Inner product space Bases and frames in Hilbert spaces Transformation skills of sums and integrals Fourier series Pointwise convergence Average convergence Several forms of the Fourier series Some problems about the Fourier series Fourier transforms Fourier transforms and inverse transforms Properties of the Fourier transforms Comparison of the Fourier series and the transforms Time - frequency window Window Fourier transforms Different definitions of the Fourier transforms and multivariate Fourier transforms Sampling theorem and filtering Sampling theorem Mathematical representations of filtering Filtering by use of sampling Filters and their response functions Associations about filters 2 Wavelet transform and its applications Wavelet transforms Univariate wavelet transforms The meanings of wavelet transforms Multivariate wavelet transforms Applications of wavelet transforms To use wavelet transforms in filtering Boundary drawing by using wavelet transforms 3 Multiresolution and orthogonal wavelets Multiresolution analysis Subspace Vj and multiresolution analysis Scaling functions Refinement relations Subspace Wj Wavelet Filter response functions and their applications Filter response functions H, G The proof of the main theorem in wavelet analysis A method to construct wavelets Effects of filters - Mallat algorithms Low pass and high pass Decomposition algorithms Reconstruction algorithms Regularity and vanishing moments of wavelets Regularity Vanishing moments Relationship of regularities and vanishing moments Wavelets are telescopes and microscopes in mathematics 4 Compactly supported real wavelets Some relative questions Forms of finite-sum refinement relations Approximate calculations Constructions of compactly supported wavelets Relative prepare knowledge Constructions of real-valued wavelets Decomposition and reconstruction algorithms Mallat algorithms Sampling algorithms Wavelet packet analysis Wavelet packet decompositions {un}and space Decompositions of Wj, Vj The equal-dilatation property Wavelet storehouses Choices of the best wavelet packet bases Best wavelet packet bases Cost functions Method to choose the best wavelet packet bases Algorithms Decomposition algorithms Reconstruction algorithms 6 Multivariate wavelets Principles of multivariate wavelets to deal with problems Bivariate multiresolution analysis Mallat algorithms Sampling algorithms 7 Biorthogonal wavelets Constructions and algorithms Constructions of biorthogonal wavelets Algorithms of biorthogonal wavelets Compactly supported biorthogonal real wavelets Compactly supported biorthogonal real wavelets Symmetric or antisymmetric real wavelets 8 Spline wavelets Simple introductions of splines Splines of order m B-splines of order m Constructions of spline wavelets Non-compactly supported orthogonal spline wavelets Compactly supported semiorthogonal symmetric or antisymmetric spline wavelets Non-compactly supported semiorthogonal symmetric or antisymmetric spline wavelets Single compactly supported biorthogonal symmetric or antisymmetric spline wavelets 9 The lifting theory of biorthogonal wavelets Principles of the lifting theory of biorthogonal wavelets Algorithms of lifted biorthogonal wavelets Examples of choices of lifting filters A direct method to construct biorthogonal wavelets Bibliography